What if objects had different masses?

Roll a ball into another ball. Bump! The second ball rolls away.

But what if the second ball were very, very heavy?

Things bump into each other all the time: marbles, balls on a pool table, hockey pucks on ice. What happens after the bump depends a lot on how heavy each thing is.

In this lab, an orange puck slides along a smooth track and hits a blue puck that is sitting still. First they weigh the same. Then the blue puck becomes ten times heavier. Can you guess how the bump will change?

A collision is a brief, strong interaction in which two objects exchange momentum. The outcome depends on the masses, the velocities and how springy the contact is.

This lab fixes the springiness (a perfectly elastic hit) and the approach speed (2 m/s), and asks one question: how do the final velocities change when the target's mass goes from equal to the projectile's to ten times larger?

First, both pucks weigh the same. The orange puck hits the blue one.

The orange puck stops. The blue puck slides away just as fast!

When the two pucks weigh the same, something neat happens. The orange puck slides in at 2 meters per second, hits the blue puck and stops dead. The blue puck slides away at 2 meters per second.

All the motion has passed from one puck to the other. Because nothing was lost, scientists call this a perfectly bouncy, or elastic, collision.

With m₁ = m₂ and a perfectly elastic head-on hit, the formulas give v₁′ = 0 and v₂′ = v₁: the pucks exchange velocities. Momentum (2 kg·m/s) and kinetic energy (2 J) are both unchanged.

This exchange is an idealized result: the model's pucks are smooth, do not spin and feel no friction. OpenStax notes that real large-scale collisions are never quite elastic, because some kinetic energy always becomes heat and sound.

Now the blue puck is ten times heavier. The orange puck hits it.

Boing! The orange puck bounces back. The heavy puck only moves a tiny bit.

Now make the blue puck ten times heavier. The orange puck hits it at the same speed, but this time it bounces backward at about 1.6 meters per second. The heavy blue puck only creeps forward at about 0.4 meters per second.

Now try it the other way: make the orange puck heavy and the blue one light. The heavy puck keeps going, and the light one shoots off faster than the heavy one was moving.

With m₂ = 10 kg, the model gives v₁′ = (1 − 10)(2)/11 ≈ −1.64 m/s and v₂′ = 2(1)(2)/11 ≈ 0.36 m/s. The negative sign means the light puck rebounds.

Check the bookkeeping: momentum after is 1(−1.64) + 10(0.36) = 2.0 kg·m/s, as before. Swapping the masses (10 kg hitting 1 kg) gives 1.64 m/s and 3.64 m/s: the heavy puck barely slows and the light one leaves at nearly twice the original speed.

In a bump, both pucks push on each other just as hard.

A heavy puck is hard to get moving. So the light puck gets pushed back instead.

During the bump, the two pucks push on each other equally hard. That push changes the light puck's motion a lot but the heavy puck's motion only a little, because heavy things are harder to speed up or slow down.

Scientists keep track of this with momentum: mass times speed. The total momentum of both pucks is the same before and after every bump. Our model's pucks are perfectly smooth and bouncy, which real objects never quite are.

Newton's third law gives equal and opposite impulses, so the total momentum m₁v₁ + m₂v₂ is conserved. An elastic collision also conserves kinetic energy, and the two conditions together fix both final velocities.

The model assumes smooth, non-spinning pucks with no friction, an instantaneous contact and a constant coefficient of restitution e. Set e below 1 and some motion energy turns into heat and sound, but momentum is still conserved. Real restitution also changes with impact speed.

Real bumps are never quite perfectly bouncy. A little motion turns into heat and sound.

On ice, two steel blocks bump and bounce apart almost perfectly.

Real bumps always turn a little motion energy into heat and sound, so they are never perfectly bouncy. Some come close. Physics books describe two steel blocks bumping on ice, or two carts with spring bumpers on an air track, where a cushion of air takes away most of the friction.

Physics books show the momentum rule with billiard balls and ice hockey pucks too. However the balls fly apart, the total momentum after the hit matches the total before.

OpenStax gives steel blocks colliding on ice and spring-bumpered carts on an air track as examples of nearly elastic macroscopic collisions; icy surfaces and air tracks are nearly frictionless. Every macroscopic collision still converts some kinetic energy into heat and sound.

The model's formulas also show the limits: as m₂/m₁ grows without bound, v₁′ → −v₁ and v₂′ → 0, so the light puck rebounds at its incoming speed; for m₁ ≫ m₂, the target leaves at nearly 2v₁. Momentum conservation itself holds for any collision, including glancing ones between billiard balls.